L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (1 − i)5-s − 1.00i·6-s + i·7-s + (−0.707 − 0.707i)8-s − 1.41i·10-s + (−0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s − 1.41i·15-s − 1.00·16-s + 1.41i·17-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (1 − i)5-s − 1.00i·6-s + i·7-s + (−0.707 − 0.707i)8-s − 1.41i·10-s + (−0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s − 1.41i·15-s − 1.00·16-s + 1.41i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.106336157\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.106336157\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 5 | \( 1 + (-1 + i)T - iT^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.247261622440139756501475667614, −8.854451414401152330462355578436, −8.151441738308896635768557025892, −6.82943484862504654329666882188, −5.87137085723434943514964668129, −5.33437070949352691080574912656, −4.46618380991399483112262031522, −3.00618832973184969418913751123, −2.19380594818416302294940613023, −1.56434091456530975299482784669,
2.50629501877442638356920160121, 3.02159374548236354690454305854, 4.05196482865083162746970982279, 4.95854663777116801066667551966, 5.76275352996241568223935914065, 6.92843697485566845416744434069, 7.27579033264089885136916311457, 8.138320936612692845688248735310, 9.367625917308823434728139959305, 9.875119109785829971056868098610